article

Free access

Globally convergent polynomial iterative zero-finding using APL

Author:

Tien Chi ChenAuthors Info & Claims

ACM SIGAPL APL Quote Quad, Volume 23, Issue 1

Pages 52 - 59

https://doi.org/10.1145/144052.144079

Published: 15 July 1992 Publication History

PDF eReader

Abstract

We report the APL study of a global iteration theory for polynomial zero-finding, with the potential to converge from almost any point in the complex plane, as distinguished from classical, “neighborhood” theory requiring the starting guess to be close to the zero being sought. The iteration functions studied involve the function and its first two derivatives, evaluated at the guess Z_k.

Using the symmetric cluster as a reference, we discovered Newton's method to be unsuitable, and the multiplicity-adjusted Laguerre formula deficient. But a new formula converges to symetric cluster zeros in just one iteration excepting for roundoff error, possesses cubic neighborhood convergence, and appears reliable for general polynomials. Roundoff errors are reduced through reasonable starting guesses. Reboundingby subclusters, a major cause of nonconvergence is resolved by a local polynomial approach. Most polynomials tested converge within a dozen iterations.

References

[1]

Kendall E. Atkinson, An Introduction to Numerical Analysis, Wiley (New York 1978). P. 87.

Google Scholar

[2]

Tien Chi Chen, "Global convergence to zeros in the presence of clusters," Proc. Inter-national __~_m_~., Taipei, Taiwan, pp. 270-276, 1988.

Google Scholar

[3]

Tien Chi Chen, "Iterative zero-finding revisited," pp. 583-590 in W.L. Hogarth and B.J. Noye (Eds.), Computational Techniques and Ap- plications: CTAC-89 (Proc. Computational Techniques and Applications Conference, Brisbane, Australia, July 1989), Hemisphere Pub. Corp. (New York 1990).

Google Scholar

[4]

E. Hansen and M. Patrick, "Estimating the multiplicity of a root," Numerische Math., vol. 27, pp. 121-131 (1976).

Digital Library

Google Scholar

[5]

E. Hansen and M. Patrick, "A family of root finding methods," Numerische Math., vol. 27, pp. 257-269 (1977).

Digital Library

Google Scholar

[6]

A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis, 2nd. Ed., McGraw-Hill (New York 1978). Pp.391, 395.

Google Scholar

Cited By

View all

(2007)Newton's and Related MethodsNumerical Methods for Roots of Polynomials, Part I10.1016/S1570-579X(07)80008-5(131-206)Online publication date: 2007
https://doi.org/10.1016/S1570-579X(07)80008-5
Chen TLuk W(1994)Aberth's method for the parallel iterative finding of polynomial zerosACM SIGAPL APL Quote Quad10.1145/190468.19028125:1(40-49)Online publication date: 1-Aug-1994
https://dl.acm.org/doi/10.1145/190468.190281
Chen TLuk W(1994)Aberth's method for the parallel iterative finding of polynomial zerosProceedings of the international conference on APL : the language and its applications: the language and its applications10.1145/190271.190281(40-49)Online publication date: 1-Aug-1994
https://dl.acm.org/doi/10.1145/190271.190281
Show More Cited By

Index Terms

Globally convergent polynomial iterative zero-finding using APL

Recommendations

Globally convergent polynomial iterative zero-finding using APL
APL '92: Proceedings of the international conference on APL

We report the APL study of a global iteration theory for polynomial zero-finding, with the potential to converge from almost any point in the complex plane, as distinguished from classical, “neighborhood” theory requiring the starting guess to be close ...
A New QP-Free, Globally Convergent, Locally Superlinearly Convergent Algorithm For Inequality Constrained Optimization

In this paper, we propose a new QP-free method, which ensures the feasibility of all iterates, for inequality constrained optimization. The method is based on a nonsmooth equation reformulation of the KKT optimality condition, by using the Fischer--...
Globally convergent Jacobian smoothing inexact Newton methods for NCP

A new smoothing algorithm for the solution of nonlinear complementarity problems (NCP) is introduced in this paper. It is based on semismooth equation reformulation of NCP by Fischer---Burmeister function and its related smooth approximation. In each ...

Comments

Information & Contributors

Information

Published In

ACM SIGAPL APL Quote Quad Volume 23, Issue 1

July 1992

309 pages

ISSN:0163-6006

DOI:10.1145/144052

Editor:
Lynne C. Shaw

Issue’s Table of Contents

APL '92: Proceedings of the international conference on APL
July 1992
326 pages
ISBN:0897914775
DOI:10.1145/144045
Editor:
Lynne C. Shaw

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 15 July 1992

Published in SIGAPL Volume 23, Issue 1

Check for updates

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
298
Total Downloads

Downloads (Last 12 months)36
Downloads (Last 6 weeks)10

Reflects downloads up to 09 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

(2007)Newton's and Related MethodsNumerical Methods for Roots of Polynomials, Part I10.1016/S1570-579X(07)80008-5(131-206)Online publication date: 2007
https://doi.org/10.1016/S1570-579X(07)80008-5
Chen TLuk W(1994)Aberth's method for the parallel iterative finding of polynomial zerosACM SIGAPL APL Quote Quad10.1145/190468.19028125:1(40-49)Online publication date: 1-Aug-1994
https://dl.acm.org/doi/10.1145/190468.190281
Chen TLuk W(1994)Aberth's method for the parallel iterative finding of polynomial zerosProceedings of the international conference on APL : the language and its applications: the language and its applications10.1145/190271.190281(40-49)Online publication date: 1-Aug-1994
https://dl.acm.org/doi/10.1145/190271.190281
Chen T(1993)SCARFS, an efficient polynomial zero-finder systemACM SIGAPL APL Quote Quad10.1145/166198.16620424:1(47-54)Online publication date: 1-Sep-1993
https://dl.acm.org/doi/10.1145/166198.166204
Chen T(1993)SCARFS, an efficient polynomial zero-finder systemProceedings of the international conference on APL10.1145/166197.166204(47-54)Online publication date: 1-Sep-1993
https://dl.acm.org/doi/10.1145/166197.166204

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Abstract

References

Cited By

Index Terms

Recommendations